Lesson | Loading...

Unit 6, Lesson 9

Introduction to Trigonometric Functions

$Course Icon$

Integrated Math 3

9.1

Warm-up

Standards Alignment

Building On

Addressing

Building Toward

Instructional Routines

None

Materials

None

Activity Narrative

The purpose of this Warm-up is to prepare students for “unwrapping” the unit circle in the next activity and seeing (for the first time in this unit) what the graphs of and look like. Part of thinking of cosine and sine as functions is understanding their input and output. The prompts in this activity ask students to informally describe the output of the two trigonometric functions, by focusing on each coordinate of a point on the unit circle.

Launch

Display the image for all to see.

Activity

None

Suppose there is a point at on the unit circle.

<p>A circle with center at the origin of an x y plane. Point P lies on the outside of the circle, on the x axis, to the right of the origin.</p>

Describe how the -coordinate of changes as it rotates once counterclockwise around the circle.
Describe how the -coordinate of changes as it rotates once counterclockwise around the circle.

Activity Synthesis

Invite students to share their descriptions, highlighting responses that use actual values such as 1, 0, and -1.

9.2

Activity

Instructional Routines

Extend It MLR8: Discussion Supports

Materials

To Gather

Graphing technology

Activity Narrative

The goal of this activity is for students to plot the relationship between the values of the - and -coordinates of points on the unit circle and their corresponding angles from 0 to . Sometimes known as “unwrapping” the unit circle, students first focus on the -coordinate and generate a graph of cosine. They then do the same for the -coordinate and generate a graph of the sine function. In this way, students use the structure of a unit circle to graph periodic functions (MP7).

In order to plot the points, students should either reference previously made displays of the unit circle, in which the approximate coordinates for the points are listed, or use technology to approximate the values of cosine and sine.

Monitor for any students who make the connection between the general shape of the graphs and the descriptions from the Warm-up, to share during the whole-class discussion.

9.3

Activity

Instructional Routines

Graph It MLR2: Collect and Display

Materials

To Gather

Graphing technology

Activity Narrative

Now that the functions and have been introduced, the goal of this activity is for students to practice graphing these functions using technology. This informal exploration of the graphs gives students opportunities to make connections to their work with the unit circle, to the Pythagorean Identity, and to an earlier unit in which they combined function types using arithmetic operations.

Monitor for students who use clear reasoning for the values of , where , to share during the whole-class discussion—for example, by thinking about the points on the unit circle where or by recalling earlier work with tangents.

Lesson Synthesis

Tell students to close their books or devices, and then display this graph:

<p>Graph of a trigonometric function</p>

Tell students that this is a graph of . Here are some questions for discussion:

“What is the value of the second tick mark on the -axis?” ( because this is where the sine function takes its maximum value of 1.)
“What is the first tick mark?” ( because it is halfway between 0 and .)
“What is the last tick mark?” ( because that is a full circle and when the sine function completes its cycle of values. is the period of the sine function.)
“Why does the graph cross the axis at the 4th tick mark?” (That crossing corresponds to on the unit circle, which is when the height of the point on the circle goes from above the -axis to below the -axis as increases.)

to access Cool-down.

Student Lesson Summary

Using the unit circle, we can make sense of and for any angle measure between 0 and radians. For an angle , starting at the positive -axis, there is a point, , where the terminal ray of the angle intersects the unit circle. The coordinates of that point are .

<p>A circle with center, point B, at the origin of an x y plane.</p>

But what if we wanted to think about just the horizontal position of point as goes from 0 to ? The horizontal location is defined by the -coordinate, which is . If we graph , we get:

This graph is 1 when is 0 because the -coordinate of the point at 0 radians on the unit circle is . The graph then decreases to -1 (the smallest -value on the unit circle) before increasing back to 1.

We can do the same for the -coordinate of a point on the unit circle by graphing :

This graph is 0 when is 0, increases to 1 (the greatest -value on the unit circle), then decreases to -1 before returning to 0.

Lesson | Loading...

Settings

Audience

Assessment Access

Lesson 9

Introduction to Trigonometric Functions

Warm-up

Standards Alignment

Building On

Addressing

HSF-TF.A

Building Toward

HSF-IF.C.7

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Activity Synthesis

Activity

Instructional Routines

Materials

To Gather

Activity Narrative

Activity

Instructional Routines

Materials

To Gather

Activity Narrative

Lesson Synthesis

Student Lesson Summary

Student Response

Building on Student Thinking

Launch

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Launch

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Standards Alignment

Building On

Addressing

HSF-IF.C.7

Building Toward

Standards Alignment

Building On

HSF-BF.A.1.b

Addressing

HSF-IF.C.7

Building Toward